chameleon Twistor Theory: a Geometric Programme for Describing the Physical World.

by Roger Penrose

Mathematical Institute, Oxford

Abstract Original motivations are recalled, for the introduction twistor theory, as a distinctive complex-geometric approach to

the basic physics of our world, these being aimed at applying specifically to (3+1)-dimensional space-time, but where space-

time itself is regarded as a notion secondary to the twistor geometry and its algebra. Twistors themselves may be initially

pictured as light rays—with a twisting aspect to them related to angular momentum. Twistor theory provides an economical

conformally invariant description of quantum wave functions for massless particles and fields, best understood in terms of

holomorphic sheaf cohomology, subsequently leading to a non-linear description of anti-self-dual (“left-handed”) gravitational

(and Yang-mills) fields. Attempts to remove this anti-self-dual restriction (the googly problem) led to a 40-year blockage to

the development of twistor theory as a possible overall approach to fundamental physics. However, in recent years, a more

sophisticated approach to this problem has been developed—referred to as palatial twistor theory—whose basic procedures

are described here, where a novel generating-function approach to -vacuum Einstein equations is introduced.

CONTENTS

Part A. Early motivations

A1. Geometrical background: two roles for a Riemann sphere

A2. The 2-spinor formalism

A3. Zero rest-mass fields

Part B. The emergence of twistor theory

B1. Robinson congruences

B2. Twistors in terms of 2-spinors

B3. Minkowski space compactified, complexified, and its conformal symmetry

B4. The basic twistor spaces

B5. Helicity and relativistic angular momentum

B6. Description under shift of origin

Part C: Fields, quantization and curved space-time

C1. Twistor quantization rules

C2. Twistor wave functions

C3. Twistor generation of massless fields and wave functions

C4. Singularity structure for twistor wave functions

C5. Čech cohomology

C6. Infinity twistors and Einstein’s equations

Part D: Palatial twistor theory

D1. Basic ideas of palatial twistor theory

D2. The spaces of momentum-scaled and spinor-scaled rays

D3. A palatial role for geometric quantization

D4. Palatial generating functions and Einstein’s equations

Part A. Early motivations

A1. Geometrical background: two roles for a Riemann sphere

The basic geometrical proposal underlying twistor theory effectively came together in

early December 1963, when I was on a 9-month appointment at the University of Texas in

Austin [1]. Various motivational notions had been troubling me for several years previously,

concerning what I had felt to be a need for a novel approach to foundational physics, in which

concepts from both quantum mechanics and relativity theory had significant roles to play.

These were interrelated via the theme of complex analysis and complex-number geometry,

areas of mathematics that had impressed me deeply from around 1950, during my time as an

undergraduate in mathematics at University College, London. These ideas had then featured

strongly in my mind in the early 1960s. The thought I had in late 1963 was the initial stage of

the proposal that, a little later, I indeed referred to as “twistor theory”, owing to a key role that

the twisted configuration of interlocking circles shown in figure 1 (a stereographically

projected family of the Clifford parallels on a 3-sphere) had played for me. The reader might

well ask what such an intriguing configuration might have to do with a basic theory of physics.

We shall see later that this configuration represents the angular momentum of a massless

particle with spin, but in order to explain this, it is necessary first to outline some of the various

ideas that had been troubling me earlier. I shall come to the specific role of the configuration

of Fig.1 in §B1, §B4. and B5, particularly t the end of that section.

Fig. 1:

A picture representing a non-null twistor: stereographic projection—to a Euclidean 3-space E—of Clifford

parallels on a 3-sphere. The tangent directions to the circles point in the direction (projected into E) of the rays of

a Robinson congruence. By continually reassembling itself, the entire configuration travels with the speed of light,

as E evolves in time, in the direction of the large arrow at the top right. The configuration represents the angular

momentum structure of a massless particle with spin.

One of my main motivations had arisen from my feeling that there was a need for a

formalism that was geared to that specific dimensionality of space-time structure that we

directly perceive around us. This line of thinking was very unlike that of various other ideas

for an underlying physics of the world that later became popular, e.g. string theory [2]. I had

earlier become convinced that what was needed would be a formalism that should be very

specific to the number of space and time dimensions, namely 3 and 1, respectively, that

macroscopically present themselves to us, and I took the view that this should be central to the

scheme. This indeed goes very much in opposition to the role of space-time dimensionality

underlying many of the current trends, most particularly string theory, where extra space

dimensions (and even an extra time dimension, in the case of “F-theory”) are regarded as

essential ingredients of these various theories [2], taken to be serious proposals for the overall

space-time geometry of the physical world that we inhabit. It also contrasts with the very

natural and commendable desire, in pure mathematics, for formalisms that can be applied,

generally, to any spatial dimensionality whatever, but the aims of theoretical physics are very

different from those of pure mathematics, even though much of theoretical physics depends

vitally on the latter.

Another of my basic motivations had been for a formalism that was essentially complex

in the sense that it would be able to take advantage of what I had regarded, ever since my days

as a mathematics undergraduate, as the “magic” of complex analysis and holomorphic (i.e.

complex analytic) geometry. I had learnt that the complex number system has not only a

profoundly deep power and elegance, but that it had also found a basic realization in its

underlying role in the formalism of quantum theory. I later began to study quantum mechanics

in a serious way, and was particularly impressed by the superb course of lectures given by Paul

Dirac, when I was a graduate student (in algebraic geometry), and subsequently a Research

Fellow, at St John’s College Cambridge. I became fascinated by the quantum description of

spin, and how the complex numbers of quantum mechanics were directly related to the 3-

dimensiality of physical space, via the 2-sphere of spatial directions being appropriately

identified as a Riemann (or Bloch) sphere of the ratios of pairs of complex numbers (quantum

amplitudes) where, in the case of a massive particle of spin ½ such as an electron (see figure

2), we can think of these as being the complex components of a 2-spinor. Moreover, I had

realized that in the relativistic context, there was another role for the Riemann sphere, this time

as the celestial sphere that an astronaut in space would observe. The transformation of this

celestial sphere to that of a second astronaut, moving at a relativistic speed while passing

nearby the first would be one that preserves the complex structure of the Riemann sphere (i.e.

conformal without reflection). The special (i.e. non-reflective) Lorentz group is thus seen to be

identical with these holomorphic transformations of this Riemann sphere (Mobius

transformations). Again this was clear from the 2-spinor formalism, this time in the relativistic

context (see [3]).

Fig. 2:

The Riemann sphere (here in its role as a Bloch sphere) projects stereographically from its south pole S to the

complex (Wessel) plane, whose unit circle coincides with the equator of the sphere. A general spin state |↗

=w|↑+z|↓, of a spin-½ massive particle is represented by the pint Z on the Wessel plane denoting the complex

number u=z/w, which is the stereographic image of Z´ on the sphere (so S, Z, and Z´ are collinear). The spin

direction ↗ is then OZ, where O is the sphere’s center.

A2. The 2-spinor formalism

This dual role for the Riemann sphere, one fundamentally to do with quantum

mechanics in the case of 3 spatial dimensions, and the other fundamentally to do with

macroscopic relativity, in (3+1)-dimensional space-time, struck me as being no accident, but

something that linked together these two great revolutions of 20th century physics—of the small

and of the large—via the magic of complex numbers. I felt that this might represent a definite

clue to a deep unifying relation between the two. Both could be seen as a feature of the 2-spinor

calculus, as introduced be Cartan [4] and van der Waerden [5], and which I had learnt how to

use from Dirac (see [6]), in an unexpected deviation from his normal Cambridge course on

quantum mechanics.

I liked to think of a 2-spinor (often referred to by physicists as a “Weyl spinor”) in a

very geometrical way, and I realized that, up to an overall sign, a non-zero 2-spinor can be

represented as a future-pointing null vector (a vector pointing along the future null cone),

referred to as the “flagpole”, together with a “flag plane” direction through that flagpole [7],

[8]. The flag plane would be a null half-plane bounded by the flagpole. This flag geometry can

be thought of in the following way. Imagine the Riemann sphere 𝒮 of null (i.e. lightlike)

directions at some point O in space-time. (See figure 3.) We are thinking of the geometry in

the tangent 4-space of the point O. The flagpole direction is represented by some point P on a

sphere of cross-section of the future null cone of O, which we identify with 𝒮, and we choose

a point P´ on 𝒮 infinitesimally separated from P. The straight line extended out from P in the

direction of P´, when joined to O, defines the required flag half-plane. We note that as the point P´

rotates about P, the flag plane rotates about the flagpole. The spinor itself is defined only up to

Riemann sphere

N

S

Z u O i

–i 1

–1

Z '

imaginary axis

real axis

sign by this geometry, but we must take note that if P´ rotates continuously around P through

2, the spinor becomes replaced by its negative. To reach the original 2-spinor by this

procedure, the rotation of the flag plane would have to be through 4.

Fig. 3:

(a) The space of null directions at some space-time point O is represented as a Riemann 2-sphere 𝒮. The flagpole

direction of a 2-spinor is represented, on 𝒮, as the point P. Infinitesimally near to P is P´, where the direction 𝑃𝑃′⃗⃗ ⃗⃗ ⃗⃗ provides the 2-spinor’s flag plane.

(b) In space-time terms, the 2-spnor’s flagpole is shown as the null 4-vector 𝑂𝐹⃗⃗⃗⃗ ⃗ , where we realize 𝒮 as a particular

3-plane intersection of the future null cone of O (all this taken in O’s tangent 4-space), so that P lies on the line

OF. The 2-spinor’s flag plane is now seen as the null half-2-plane extending away from the line OF in the direction

of P´.

I had found that 2-spinor methods were surprisingly valuable in giving us insights into

the formalism of general relativity that were different from those that the standard Lorentzian

tensor framework readily provides. Most immediately striking was the very simple-looking 2-

spinor expression for Weyl’s conformal curvature [9] (see also [10]). Whereas the usual Weyl-

tensor quantity Cabcd, has a somewhat complicated collection of symmetry and trace-free

conditions, the corresponding 2-spinor is simply a totally symmetric complex 2-spinor quantity

ABCD.

Some comments concerning the 2-spinor index notation being used here are

appropriate. Capital italic Latin index letters A, B, C, … refer to the (2-complex dimensional)

spin space if they are upper indices, and to the dual of this space if lower ones; primed such

letters A´, B´, C´, … refer to the complex-conjugate spin space. The tensor product of the spin

space with its complex conjugate is identified with the complexified tangent space to the space-

time, at each of its points, here the real tangent vectors arise as the Hermitian members of this

tensor product. In general, I shall take these as abstract indices, in the sense described in my

book with Wolfgang Rindler, Spinors and Space-Time, volume 1 [8], so that no coordinate

system is implied, either for the space-time or to define a basis for the spin-space. This is

notationally very handy, because the space-time indices a, b, c, … can then be thought of as

“shorthand” for the spinor index pairs:

a = AA´, b = BB´, c = CC´, …

The spin-space (and hence also its dual and complex conjugate) has a symplectic structure

defined by the skew-symmetric quantities

AB, AB, A´B´, A´B´,

these being used for lowering or raising indices, (where we must be a little careful about signs

and index orderings):

B = A AB, A = B AB, B´ = A´ A´B´, A´ = B´ A´B´

( a ) ( b )

P

O

P

F

so that on terms of components,

1 = 0, 0 = –1, 1´ = 0´, 0´ = –1´,

where the component form of each of the epsilons is

(0 1

−1 0).

The metric tensor, in abstract-index form is

gab = AB A´B´,

and the abstract-index form of the Weyl conformal curvature tensor for space-time is

Cabcd = ABCD A´B´ C´D´ + AB CD �̃�A´B´C´D´.

Here, I have allowed for the case of a complex metric gab, both ABCD and Ψ̃A´B´C´D´ being totally

symmetric, where ABCD describes the anti-self-dual (left-handed) Weyl curvature and Ψ̃A´B´C´D,

the self-dual (right-handed) part. In the case of a real Lorentzian space-time metric (휀A̅B=AB)

and Ψ̃A´B´C´D´ is the complex conjugate of ABCD:

�̃�A´B´C´D´ = �̅�A´B´C´D´,

but it will be important for what follows that we consider the complex case also, as we shall be

concerned with self-dual (complex vacuum) space-times, for which ABCD=0 and anti-self-dual

ones, for which �̃�A´B´C´D =0, later (these complex fields being regarded as wave functions).

A3. Zero rest-mass fields

We find that in the case of a (real Lorentzian) vacuum metric (with or without

cosmological constant), the Bianchi identities become

AA´ABCD = 0

which may be compared with the Maxwell equations in charge-free space-time

AA´AB = 0,

where AB relates to a (possibly complex) Maxwell field tensor Fab in the same way as ABCD

relates to Cabcd, namely

Fab = AB A´B´ + AB �̃�A´B´,

where AB describes the anti-self-dual (left-handed) part of the field and �̃�A´B´, the self-dual

(right-handed) part. For a real Maxwell field, they are complex conjugates of each other:

�̃�A´B´ = �̅�A´B´.

I had become interested in the issue of finding solutions of the general equation

AA´ABC….E = o

in (conformally) flat space-time, ABC….E being symmetric in its n spinor indices, the equation

being the (conformally invariant) free-field equation for a massless field of spin n/2 [6], [11],

[12]. This equation (together with the wave equation in suitably conformally invariant form,

which includes an R/6 term, R being the scalar curvature) had a particular importance for me,

and I believed it to have a rather basic status in relativistic physics. For I had come to the view

that nature might have a “massless’ structure at its roots, mass itself being a secondary

phenomenon. In around 1961 (see [13]) I had found a formula for obtaining the solution of this

field equation from general data freely specified on a null initial hypersurface. I had formed

the view that this formula had a certain kinship with the Cauchy integral formula for obtaining

the value of a holomorphic function at some point of the complex plane in terms of the

function’s values along a closed contour surrounding that point. I had felt that, in some sense,

this massless field equation might be akin to the Cauchy-Riemann equations. There had to be

in some unusual “complex” way of looking at Minkowski space, I had surmised, in which the

massless field equations were simply a statement of holomorphicity—but in what sense could

this possibly be true?

There was one remaining feature that I felt sure must be represented, as part of this

mysterious “complex” way of looking at space-tine. This arose from a discussion that I had

had with Engelbert Schücking when I shared an office with him in the spring of 1961 at

Syracuse University in New York State. Engelbert had persuaded me of the key importance to

quantum field theory of the splitting of field amplitudes into positive and negative frequency

parts. I was not happy with the standard procedure of first resolving these amplitudes into

Fourier components and then selecting the positive ones, as not only did this strike me as too

“top-heavy”, but also the Fourier analysis is not conformally invariant—and I had come to

believe that this conformal invariance, being a feature of massless fields, was important (again,

something that had been stressed to me by Engelbert).

I had become aware that for complex functions defined on a line (thought of as the time

line) we may understand their splitting into positive- and negative-frequency parts in the

following way. We view this time line as being the equator of real numbers in a Riemann

sphere which, as before, is the complex plane compactified by the single point labelled by “∞”,

but where the sphere is now being oriented somewhat differently from that of figure 2, with

the real numbers now featuring as the equator (increasing as we proceed in an anti-clockwise

sense un the horizontal plane), rather than the unit circle. Functions defined on this equatorial

circle which extend holomorphically into the southern hemisphere (with usual conventions)

are the functions of positive frequency, and those which extend holomorphically into the

northern hemisphere are those of negative frequency. An arbitrary complex function defined

on this circle can be split into a function extending globally into the southern hemisphere and

one globally into the northern hemisphere—uniquely except for an ambiguity with regard to

the constant part—and this provides us with the required positive/negative frequency split,

without any resort to Fourier analysis. I wanted to extend this picture into something more

global, with regard to space-time, and I had in mind that my sought-for “complex” way of

looking at Minkowski space should exhibit something strongly analogous to this division into

two halves, where the boundary between the two could be interpreted in “real” terms, in some

direct way. This had set the stage for the emergence of twistor theory.

Part B. The emergence of twistor theory

B1. Robinson congruences

A colleague of mine, Ivor Robinson, who had taken up a position at what later became the

University of Texas at Dallas, had been working on finding global non-singular null solutions

of Maxwell’s free-field equations in Minkowski space-time 𝕄, where “null” in this context

means that the invariants of the field tensor Fab vanish, i.e. FabFab=0=*FabF

ab where *Fab is the

Hodge dual of Fab. Equivalently, in 2-spinor terms, ABAB=0, which tells us that

AB = AB,

for some A. It is not hard to show that the Maxwell source-free equations then imply that the

flagpole direction of A points along a 3-parrameter family—a congruence—of null straight

lines, which turn out to be what is called “shear-free”, which means that although the lines may

diverge, converge, or rotate, locally, there is no shear (or distortion) as we follow along the

lines.

Although, not relevant to the discussion at the moment, it is worth noting that the study

of shear-free congruences of rays in curved space-times has a considerable historical

significance—where I use the term “ray” simply to mean a null (i.e. lightlike) geodesic in

space-time. In particular, the well-known Kerr solution [14], [15] of the Einstein vacuum

equations for a rotating black hole possesses a shear-free ray congruence, and this played a key

role in its discovery, as it did also in Newman’s generalization to an electrically charged black

hole [16], and also in the Robinson-Trautman gravitationally radiating exact solutions [17],

among other examples. As in the case of Minkowski space 𝕄, as described above, it is also

true that for any null solution AB of Maxwell’s equations in curved space-times, the flagpole

directions of the A-spinors point along a shear-free family of rays.

A simple example of a shear-free ray congruence in 𝕄 is obtained from any fixed

choice of a ray L in 𝕄, where the family of all rays that meet L provides a shear-free ray

congruence. I refer to such a congruence as a special Robinson congruence, and this includes

the limiting case when L is taken out to infinity, so our congruence becomes a family of parallel

rays in 𝕄. Ivor Robinson had developed ways of producing null solutions of the Maxwell

equations, starting from any given shear-free null congruence, but when applied to the special

congruences just described, he found that singularities would arise along the line L itself

(except in the otherwise unsatisfactory case where L is a is at infinity). Desiring a singularity-

free Maxwell field, he provided the following ingenious trick. Consider, instead, solutions of

Maxwell’s equations in the complexified Minkowski space-time ℂ𝕄, and displace the line L in

a complex direction, so that it lies in ℂ𝕄, but entirely outside its real part 𝕄. Complex analytic

solutions of Maxwell’s equations, based on the complex “special Robinson congruence”

defined by the displaced L need not now be singular within 𝕄, and the flagpoles of the A-

spinors within 𝕄 now point along an entirely non-singular sear-free ray congruence in 𝕄,

which I later named a (general) Robinson congruence.

I became highly intrigued by the geometry of general Robinson congruences, and I soon

realized that one could describe them in the following way. Consider an arbitrary spacelike 3-

plane E in Minkowski 4-space 𝕄. E has the geometry of ordinary Euclidean 3-space, and each

ray N of the congruence will meet E in a single point, at which we can determine the location

of that ray within 𝕄 by specifying a unit 3-vector n at that point, pointing in the spatial direction

that is the orthogonal projection into E of the null direction of N there. Thus we have a vector

field of ns within E to represent the Robinson congruence. After some thought I realized what

the nature of this vector field must be. The n-vectors are tangents to the oriented circles

(together with one oriented straight line) obtained by stereographic projection of a family of

oriented Clifford parallels on a 3-sphere. See figure 1, in §A1, for a picture of this

configuration, and reference [18] for a detailed derivation. The large arrow at the top right

indicates the direction in which the configuration appears to move with the speed of light by

continually reassembling itself in that direction, as E moves by parallel displacement into the

future.

By examining this configuration, and counting the number of degrees of freedom that

such configurations have, I realized that the space of Robinson congruences must be 6-

dimensional. Moreover, it was reasonably clear to me that by its very mode of construction,

this space ought to have a complex structure, and so must be, in a natural way, a complex 3-

manifold. Within this space would lie the space of special Robinson congruences, each of

which would be determined by a single ray (namely L). The space of rays in 𝕄 is 5-real-

dimensional, and it divides the space of general Robinson congruences into two halves, namely

those with a right-handed twist and those with a left-handed twist. The complex 3-space of

Robinson congruences, which came to be known as “projective twistor space” appeared to be

just what I believed was needed, where the “real” part of the space (representing light rays in

𝕄, or their limits at infinity) would, like the “real” equator of the Riemann sphere described at

the end of §A2, divide the entire space into two halves. This, indeed appeared to be exactly the

kind of thing that I was looking for!

B2. Twistors in terms of 2-spinors

To be more explicit about things, and to understand precisely how the space of

Robinson congruences does indeed provide a compact complex 3-manifold divided in two by

the real 5-space of special Robinson congruences, let us turn again to the relativistic 2-spinor

formalism of §A2. We shall see how this allows us to provide a very neat description of

individual rays in 𝕄. In §B4, we see how this generalizes to describe general Robinson

congruences. The physical interpretation in terms of relativistic angular momentum of

massless particles will emerge in B5.

Consider some ray Z in 𝕄, and let us assign a strength to this ray in the form of a null

4-momentum convector pa, where the vector pa points along Z at each of its points, parallel-

propagated along Z. In fact, let us go a little further than this by assigning a (dual, conjugate)

2-spinor A, parallel-propagated along Z, where

pa = �̅�𝐴𝜋𝐴′,

so that in addition to having A’s flagpole pointing along Z , we also have A’s flag plane

(and spinor sign) assigned to Z, and which is to be parallel-propagated along it. This will be

referred to as a spinor scaling for the ray Z.

We need to choose a space-time origin point O within 𝕄, so that any point X of 𝕄 can

be labelled by a position vector xa (=xAA ) at O. Then if X is any point on the ray Z, we can

define a 2-spinor A by the equation,

A = i𝑥𝐴𝐴′𝜋𝐴′

and we find that A remains unchanged if X is replaced by any other point on the ray Z, such

a point having a position vector of the form

xAA + k A ̅A,

where k is any real number (since AA=0). The pair (A, A), serves to identify the ray Z,

together with a spinor scaling for Z.

The 2-spinors A and A are the spinor parts (with respect to the origin O) of the

twistor Z, which represents the spinor-scaled ray Z, and often one simply writes

Z = (A, A).

However, for a ray, there is a particular equation that must hold between the spinor parts,

namely

A ̅A + A ̅A = 0

which follows from the fact that the vector xa is real, so that xAB has the Hermitian property

𝑥𝐴𝐵′̅̅ ̅̅ ̅̅ =xBA. The above equation can be rewritten as

𝑍�̅�𝛼 = 0

where �̅�𝛼, the complex conjugate of 𝑍

�̅�𝛼 = (̅A, ̅A ),

(and note the reverse order of the spinor parts) is a dual twistor. When 𝑍�̅�𝛼 = 0, we refer to

𝑍 as a null twistor, so it is that the null twistors represent (spinor-scaled) rays in 𝕄—or rays

at 𝕄’s infinity.

The above equation

A = i𝑥𝐴𝐴′𝜋𝐴′

is referred to as the incidence relation between the space-time point X and the twistor

Z=(A,A). We may also be interested in this incidence relation when X is allowed to be a

complex point. Likewise, for a dual twistor

Wα = (λA, A´),

incidence with a (possibly complex) point X is expressed as

A´ = −i𝑥𝐴𝐴′𝐴.

It is useful to get a picture of the geometrical role of the 2-spispinor A, in addition to

A, in the case of a general null twistor Z=(A,A). Figure4 shows this, where O is the origin

the point Q is the intersection of the ray Z with the light cone of O. The null vector OQ⃗⃗⃗⃗ ⃗ has

index form qa and is proportional to the flagpole of A where

q AA = A�̅�A (i�̅�BB)–1.

This expression fails only when �̅�BB=0 (but holding in a certain limiting sense) which occurs when

the ray Z lies in a null hyperplane through O, and the point Q lies at infinity.

Fig.4

The flagpole directions of the spinor parts of a general null twistor Z=(A,A) are depicted, where Q is

the intersection of the ray Z with the light cone of the origin O. {NEED lettering L, O, Q, , }

B3. Minkowski space compactified, complexified, and its conformal symmetry

At this juncture It would be helpful to clarify the nature of “infinity”, with regard to

Minkowski space 𝕄. We recall that when a ray L is characterized in terms of the null

congruence of rays that intersect L, we were led to consider the ray congruences that consist

entirely of parallel rays, arising when L is moved out to infinity. There is a whole 2-sphwere’s-

worth of such systems of parallel rays one for each null direction. Thus the family of limiting

rays L at infinity generates a kind of “light cone at infinity”, frequently denoted by the script

letter I (and pronounced “scri”). We can regard I as being the identification of 𝕄’s future

conformal boundary I + with its past conformal boundary I – (see [12], [18]). This

identification also incorporates the single point i (the vertex of I )e, which is the identification

of the three points i–, i0, and i+, respectively representing past infinity, spacelike infinity, and

future infinity. See figure 5. This provides us with the picture of compactified Minkowski space

𝕄 (whose turns out to have topology S1×S3) where figure 5a indicates the future and past null

boundaries of 𝕄, and figure 5b shows how these two conformal boundaries I + and I – are to

be identified as I, where future and past end-points of any ray in 𝕄 are identified. This provides

us with the highly symmetrical compact Lorentzian-conformal manifold 𝕄. Every ray within

𝕄 being compactified by a single point to become a topological circle.

(a) (b)

Fig. 5:

(a) A conformal picture indicating how Minkowski space-time 𝕄 acquires its future null boundary I +, a null 3-

surface supplying future end-points to rays in 𝕄 and, similarly, a past null boundary I – supplying past end-

points to rays in 𝕄. There are also three other conformal boundary points i+, i–, and i0 denoting future, past, and

spacelike infinity, respectively.

(b) To complete the picture of compactified Minkowski space 𝕄, we must identify I +, with I –, so that the

future end-point a+ of any ray in 𝕄 is identified with its past end-point a-. Also the three points i+, i–, and i0 must

be identified.

The global symmetry group of 𝕄 is the 15-parameter symmetry group that is

frequently referred to as the conformal group of flat 4-dimensional space-time. This group has

4 connected components since it allows for reversals of time and space orientations. I shall be

concerned here only with the connected component of the identity, referring to this group as

C(1,3).

Another way of understanding 𝕄 is that it represents the family of generator lines of

the null cone 𝒦 of the origin O2,4 (i.e. of entire rays through O2,4) in the pseudo-Minkowskian

6-space 𝕄2,4, whose signature is (+ + – – – –). See [18]. If we consider these generator lines to

be oriented, (or as being null half-lines, starting at O2,4), then we get a 2-fold cover 𝕄of 𝕄,

since an action of the pseudo-orthogonal group SO(2,4) on 𝒦 can continuously rotate any

particular one of its oriented generators into itself but with opposite orientation (i.e. reflected

in the origin O2,4). Thus, the family of oriented rays through O2,4 provides us with a realization

of the 2-fold cover 𝕄of 𝕄.

The symmetry group of the vector space 𝕋 of twistors Z =(A,A) goes a step further

than this. The pseudo-Hermitian form ||Z||=𝑍�̅�𝛼 (=A̅A+A ̅A) has split signature (+ + – –),

so the group of (complex-)linear transformations of 𝕋 that preserve the norm is the pseudo-

unitary group SU(2,2). It is, indeed, one of Cartan’s specific local isomorphisms (see [18]) that

SO(2,4) locally isomorphic to SU(2,2), the latter being a 2-fold cover of the former. This tells

us that this SU(2,2) actually acts on a 2-fold cover 𝕄of 𝕄. The space 𝕄 is therefore a 4-

fold cover of 𝕄. Topologically, we, we can understand such an n-fold cover 𝕄n, of 𝕄, as

obtained simply by “unwrapping” the S1 of 𝕄’s topology S1×S3 to the required degree n.

In fact, this strange-looking 4-fold cover 𝕄 of compactified Minkowski space can be

understood explicitly in terms of the geometrical representation of a null twistor Z in

Minkowski space-time terms. We recall that a null twistor describes not just a ray Z in 𝕄, but

also a spinor scaling, defined by A , assigned to the null direction at each point of the ray Z,

where we think of this spinor scaling as parallel-transported along the ray Z within 𝕄. Now,

we saw in §A2 (see figure 3) that a 2-spinor has a U(1) phase that is geometrically described

+

–

i +

i –

i 0

+

–

i +

i 0

i –

a +

a – identify

by a null flag half-plane, where if the flag plane is rotated about the flagpole through 2, the

spinor changes sign. However, when we think of this flag half-plane as being parallel-

transported all the way from I – to I + along Z, we find that if we were to try to match I + directly to I –, as indicated in figure 5b, then we would find a discrepancy of a rotation through

, i.e. the flags would point in the opposite directions from one another across I. (This

geometry is explained explicitly in [18], §9.4; see particularly Fig. 9-11.) Since a rotation of a

spinor flag-plane by 2 results in a change of sign for the spinor, we need 4 to get it back to

its original value. The rotation through that we find when we pass across I, represents a

discrepancy of i in the geometrical description of a null twistor in the space 𝕄. Moreover, the

problem is not removed if we consider the flag-plane interpretation within just the 2-fold cover

𝕄, since we still have a sign discrepancy. Only when we pass to the 4-fold cover 𝕄 do we

get a fully consistent picture of a null twistor—and, indeed, of a non-null twistor (see [18]),

whose interpretation we turn to next.

B4. The basic twistor spaces

Let us now consider how to represent a non-null twistor 𝑍 in a geometrical way. It is

best to think in terms of the family of null twistors Y that are orthogonal to Z in the sense

that

𝑍 �̅�𝛼 = 0

(or, equivalently 𝑌�̅�𝛼=0). If Z were a null twistor—where Y is given as a null twistor—these

respectively representing rays Z and Y, then this vanishing of their scalar product asserts that

these rays intersect (perhaps at infinity). Accordingly, if Z is fixed, then this condition on Y

tells us that the Y belongs to the special Robinson congruence defined by the ray Z. Now, let

Z be a fixed non-null twistor (but where Y remains null). Then the congruence of Y-rays subject

to orthogonality with Z will provide a general Robinson congruence. See [18] for details.

As noted in §B3, the space 𝕋 of all twistors Z is a 4-dimensional complex vector space,

with pseudo-Hermitian scalar product (𝑍�̅�𝛼) of split signature (+ + – –). Geometrical notions

are often best expressed in terms of the projective twistor space ℙ𝕋 of twistors up to

proportionality, this being a complex projective 3-space ℂℙ3. This compact complex manifold

ℙ𝕋—or, more strictly, in accordance with the above discussion, the ℂℙ3 of dual projective

twistors ℙ𝕋*—can indeed be identified with the space of Robinson congruences referred to

above. The dual twistor space 𝕋* is identified with the complex conjugate space �̅� of 𝕋 via

this pseudo-Hermitian structure. The points of the dual projective space ℙ𝕋* represent the

complex projective planes within ℙ𝕋. The complex projective lines within ℙ𝕋 correspond to

points of the complexified compactified Minkowski space ℂ𝕄.

Whereas, generally speaking, it is the projective twistor space ℙ𝕋 that is useful to us if

we are thinking of geometrical matters, the space 𝕋 is appropriate if we are concerned with the

algebra of twistors. For a non-zero twistor Zα, we can have three algebraic alternatives. These

are:

𝑍�̅�𝛼>0, for a positive or right-handed twistor Zα, belonging to the space 𝕋+,

𝑍�̅�𝛼<0, for a negative or left-handed twistor Zα, belonging to the space 𝕋–,

𝑍�̅�𝛼=0, for a null twistor Zα, belonging to the space ℕ.

The entire twistor space 𝕋 is the disjoint union of the three parts 𝕋+, 𝕋–, and ℕ, as is its

projective version ℙ𝕋 the disjoint union of the three parts ℙ𝕋+, ℙ𝕋–, and ℙℕ (see figure 6).

Fig. 6:

The way that the various parts of twistor space 𝕋 relate to their various projective counterparts of ℙ𝕋.

Each point of ℙ𝕋 represents a 1-dimenional vector subspace of 𝕋. The points of ℂ𝕄 are thus

described by 2-complex-dimensional subspaces of 𝕋 (complex lines in ℙ𝕋). The points of 𝕄

are described by 2-complex-dimensional subspaces in ℕ, i.e. by complex lines in ℙℕ. Robinson

congruences are represented by the intersections of complex projective planes in ℙ𝕋 with ℙℕ.

In the case of a special Robinson congruence, of rays meeting a particular ray L, the complex

plane in ℙ𝕋, the complex plane has contact with ℙℕ at a point representing the ray L in 𝕄.

B5. Helicity and relativistic angular momentum

It is the space ℙℕ that has the most direct physical interpretation, since its points

correspond to world-lines of free classical massless particles, which we can think of as the

classical histories of (point-like) photons in free motion, though possibly at infinity, as a

limiting case in Minkowski space-time 𝕄; see figure 7. As stated above, not only are the points

of complexified Minkowski space ℂ𝕄 represented as (complex projective) lines in ℙ𝕋, but so

also are all the points of the complexified compactified Minkowski space ℂ𝕄. Those lines

that lie in ℙℕ, represent points of the real space-time 𝕄 (possibly at infinity), but since these

lines are still complex projective lines, they are indeed Riemann spheres, in accordance with

the ambitions put forward in §A2; see figure 7.

Z

Fig. 7:

The most immediate part of the twistor correspondence: a ray Z in Minkowski space 𝕄 corresponds to a point in

ℙℕ; a point x of 𝕄 corresponds to a Riemann sphere X in ℙℕ.

In figure 8 this picture is extended to include a physical interpretation of non-null

twistors, where points of ℙ𝕋+ and ℙ𝕋– are represented, in Minkowski space, as though they are

light rays with a twist about them. This is schematic, but indeed these points can be regarded

as representing massless particles with spin. In relativistic physics, if a massless particle has a

non-zero spin, the “spin-axis” must be directed parallel or anti-parallel to the particle’s

velocity. We say that the particle has a helicity s, that can be positive or negative (or zero, for

a spinless massless particle). If s≠0, then the particle’s space-time trajectory is not precisely

defined (in a relativistically invariant way) as a world-line, but can be specified in terms of its

4-momentum pa and 6-angular momentum Mab about some chosen space-time origin point O.

These must be subject to

pa pa = 0, p0 > 0, M(ab) = 0, 1

2abcd p

b Mcd = spa

(curved or square brackets around indices respectively denoting symmetric or anti-symmetric

parts), where abcd=[abcd] is the Levi-Civita tensor fixed by its component value 0123=1 in a

right-handed orthonormal Minkowskian frame (with time-axis basis vector 𝛿0𝑎, so p0 is the

particle’s energy in units where the speed of light c=1). Note that 12abcdM

cd=*Mab is the Hodge

dual of Mab, so the second displayed equation becomes *Mabpb=spa. The connection between

these quantities and twistor theory is that if

Z = (A, A)

then we can make the interpretation

pAA´ = A´�̅�A , MAA´BB´ = i(A�̅�B) A‘B‘ – i�̅�(A‘B‘) AB,

and all the above conditions are automatically satisfied, provided that πA´≠0. Conversely, the

twistor Z (with πA´≠0) is determined, uniquely up to a phase multiplier eiθ, by pa and Mab,

subject to these conditions. The helicity s finds the very simple (and fundamental) expression

2s = ωA�̅�A + πA´�̅�A´

= Zα �̅�α.

x

Z Z

X

Riemann sphere

There is, however, the subtlety referred to above that in this interpretation of a non-null twistor,

when the helicity s is non-zero, there is no actual world-line that can describe the particle’s

location in a relativistically invariant way, the world-line being, in a sense, “spread out” in

accordance with the configuration depicted in figure 1, as we shall see at the end of §B5. This

issue is an important undercurrent to the application of twistor ideas in general relativity as

discussed in D3 and D4.

Fig. 8:

Classical massless particles with positive (right-handed) helicity can be represented as points of ℙ𝕋+ and those with negative

(left-handed) helicity, as points of ℙ𝕋–.

B6. Description under shift of origin

Under a displacement of the origin O to a new point Q of 𝕄,

O ↦ Q,

where the position vector OQ⃗⃗⃗⃗ ⃗ is (in abstract-index form) qa, the spinor parts of the twistor

Z=(A, A) undergo

ωA ↦ ωA – i qAA´πA´, πA´ ↦ πA´.

For a dual twistor Wα=(λA,A´), we correspondingly have

λA ↦ λA, A´ ↦ A´ + i qAA´ λA.

This turns out to be consistent with the standard transformation of Mab (and pa) under origin

change, where the position vector xa of a space-time point X correspondingly undergoes

xa ↦ xa – qa.

At this juncture, it is worth pointing out that whereas there may be a temptation to

identify at twistor, as represented as a pair of 2-spinors (A,A), with the 2-spinor form of a

Dirac 4-spinor (see [11], [19]), which it superficially resembles, this would be entirely

inappropriate. It is certainly true that the behavior of twistors and Dirac4-spinors is basically

the same under Lorentz transformations (leaving the origin fixed), as is defined by their spinor-

index structures. But the above behavior of a twistor under shift of origin (in effect, under

translation) has no place in Dirac’s electron theory. As we have seen in §B3, twistors provide

the basic (finite-dimensional) representation space for the (restricted) conformal group C(1,3),

whereas Dirac spinor fields provide an infinite-dimensional representation space for the

(restricted) Poincaré group. The normal Dirac equation describes a massive particle, and is not

conformally invariant. The Dirac-Weyl equation for a massless neutrino is, however, and its

relation to twistor theory is contained in the discussion given in §C3. Massive particles can

also be handled with twistor theory, but such descriptions normally require more than one

twistor. See [18], [20], [21], [22], [23]; see also the remarks given at the end of §D4.

There is a connection between the above direct physical interpretation of a twistor in

terms of angular momentum—particularly a non-null twistor Zα—and the Robinson

congruence defined by Zα. This congruence is provided by the family of rays defined by the

null (dual) twistors Wα=(λA,A), satisfying

Zα Wα = 0.

To see the connection with angular momentum, let us examine this relation at an arbitrary point

Q of 𝕄, where we now take Q as a (variable) origin point. We are interested in the ray W of

the congruence which passes through Q. With respect to Q, as origin, Wα then takes the form

Wα = (λA, 0)

(A´ being zero, since Wα is now incident with the origin point Q; see §B2, and also figure 4 in

complex-conjugate form). Accordingly, the relation ZαWα=0 now becomes

ωA λA = 0,

at the point Q. This tells us that the flagpole direction of ωA is the same as that of λA, namely

the direction of the ray W. Tus, the angular momentum Mab of the spinning massless particle

determined by Zα has a structure that is characterized by the flagpole directions of its two spinor

parts with respect to Q. We may refer back to figure 1, to see the curious spatial geometry of

all this, where the flagpole directions of ωA (small arrows in figure 1) twist around in this

complicated (Robinson congruence) way, while that of πA simply points in the direction of

motion of the configuration (large arrow at the top right of figure 1). It may perhaps be

mentioned, that the choice of letters “π” and “ω” come from the normal usage of “p”’ for

momentum, and “omega” for angular momentum.

Part C: Fields, quantization and curved space-time

C1. Twistor quantization rules

Up to this point, we have been considering twistor theory only in relation to classical

physics in flat space-time geometry. Quantum twistor theory—and, indeed, as we shall be

seeing later (in Part D), also space-time curvature—involves considering twistors (and dual

twistors) as non-commuting operators, satisfying certain commutation laws:

Zα �̅�β – �̅�β Zα = ℏ𝛿𝛽

𝛼

and, as far as our current considerations go,

Zα Z β – Z

β Zα = 0, �̅�α �̅�β – �̅�β �̅�α = 0

[18], [24]. Now, the twistors are taken to be linear operators generating a non-commutative

algebra 𝔸, whose elements are taken to be acting on an appropriate quantum “ket-space” |…

of some kind [25], but it is best not to be specific about this, just now. We could alternatively

think of our operators as dual twistors, subject to the commutation laws

Wα �̅�β – �̅�β Wα = –ℏ𝛿𝛽𝛼

and

Wα Wβ – W β Wα = 0, �̅�α �̅�β – �̅�β �̅�α = 0,

which is the same thing as before, but with �̅�α re-labelled as Wα.

These commutation laws are almost implied by the standard quantum commutators for

4-position and 4-momentum

pa xb – xb pa = iℏ𝛿𝑎

𝑏, xaxb – xbxa = 0, pa pb – pb

pa = 0,

but there appears to be an additional input related to the issue of helicity. By direct calculation,

we may verify that the twistor commutation laws reproduce exactly the (more complicated-

looking) standard commutation laws for the pa and Mab as defined in §B5, which arise from

their roles as translation and Lorentz-rotation generators of the Poincaré group (see [18]). In

this calculation, there is no factor-ordering ambiguity in the expressions for pa and Mab in terms

of the spinor parts of Zα and �̅�α (owing to the symmetry brackets). Yet, the calculation for the

helicity s (writing the operator as s) yields:

s = 14 (Zα�̅�α + �̅�αZ

α).

C2. Twistor wave-functions

If we are to express wave-functions for massless particles in twistor terms, to be in

accordance with standard quantum-mechanical procedures we need functions of Zα that are

“independent of �̅�β”. This means “annihilated by ∂/∂�̅�β”, i.e. holomorphic in Zα (Cauchy–

Riemann equations). Thus, a twistor wave-function (in the Zα-description) is holomorphic in

Zα, and the operators representing Zα and �̅�α act as:

Zα ⇝ Zα × , �̅�α ⇝ –ℏ𝜕

𝜕𝑍𝛼 ,

Alternatively, we could be thinking of functions of �̅�α that are “independent of Z β ”, i.e.

anti-holomorphic in Zα. Here it would be better to re-name �̅�α as Wα and consider functions

holomorphic in Wα. Accordingly, in the dual twistor Wα-description, a wave-function must be

holomorphic in Wα and we have the operators representing �̅�𝛼 and Wα, again satisfying the

required commutation relations, but now with:

�̅�𝛼 ⇝ ℏ𝜕

𝜕𝑊𝛼, Wα ⇝ Wα × .

To ask that our wave-function describe a (massless) particle of definite helicity, we need

to put it into an eigenstate of the helicity operator s, which, by the above, is

s = – 𝟏𝟐ℏ(Zα 𝜕

𝜕𝑍𝛼 + 2)

in the Zα-description, and

s = 𝟏𝟐ℏ(Wα

𝜕

𝜕𝑊𝛼 + 2)

in the Wα-description. These are simply displaced Euler homogeneity operators

𝚼 = Zα 𝜕

𝜕𝑍𝛼 or �̃� = Wα𝜕

𝜕𝑊𝛼,

so a helicity eigenstate, with eigenvalue s, in the Zα-description requires a holomorphic twistor

wave-function f(Zα) that is homogeneous of degree

n = –2s – 2,

where I henceforth adopt ℏ=1. Then 2s is an integer (odd for a fermion and even order for a

boson). In the Wα-description, the dual twistor wave-function 𝑓(Wα) is homogeneous of degree

ñ where

ñ = 2s – 2.

C3. Twistor generation of massless fields and wave-functions

In ordinary flat space-time terms, the position-space wave-function of a massless

particle of helicity 2s ([6], [11], [12]) satisfies a field equation, this being expressible in the 2-

spinor form

AA´AB…E = 0, = 0, or AA´�̃�A´B´…E´ = 0,

for the integer 2s satisfying s<0, s=0, or s>0, respectively, these equations having been already

considered in §A3, but where the scalar case s=0 is included also, involving the D’Alembertian

= a a.

We have total symmetry for each of the |2s|-index quantities

AB…E = (AB…E) and �̃�A´B´…E´ = �̃�(A´B´…E´).

What is the connection between the holomorphic twistor wave-function f(Zα), or dual

twistor wave-function 𝑓(Wα), with these space-time equations? In most direct terms this is

given by contour integrals [21], [24], [26], generalizing earlier expressions found by Whittaker

[27] and Bateman [28], [29]:1

AB…E(x) = k ∮𝜕

𝜕𝜔𝐴

𝜕

𝜕𝜔𝐵 …𝜕

𝜕𝜔𝐸

𝛚=i𝐱⋅𝜋 f(ω,π) π, if s≤0;

�̃�A´B´…E´(x) = k´∮ 𝜋

𝛚=i𝐱⋅𝜋 A´πB´…πE´ f(ω,π) π, if s≥0.

Here AB…E or A´B´…E´ are |2s| in number, and the 1-form π is

π = F´G´πF´ dπG´,

and where k and k´ are suitable constants. I have taken the liberty of writing xa, ωA, and πA´

without their abstract indices in places here, and using bold-face upright type instead. The

contour, for these integrals, lies within the Riemann sphere, in ℙ𝕋, of twistors Zα(ωA,πA´)

satisfying the incidence relation A=ixAAA (written ω=ix•π, below the integral sign), which

removes the ωA-dependence and introduces xa-dependence, and then the contour integration

itself removes the πA´-dependence, leaving us with just xa-dependence. Satisfaction of the field

equations is an immediate consequence of these holomorphic expressions. The 2-form

dπ0´dπ1´=1

2dπ is sometimes more appropriate to use, rather than π, the contour then being 2-

dimensional, lying in 𝕋 rather than ℙ𝕋. In the dual twistor description, we have corresponding

expressions.

See figure 8 for the geometrical set-up, where “X” is the Riemann sphere representing

the (possibly complex) point labelled xa. In this particular picture, X represents a complex pint

1 The upper one of these two displayed integral expressions was put forward by Lane Hughston [21], as a

complement to the lower one, which I had found earlier [24], [26]. The significance of having both ways of doing

it was not recognized immediately, but it was later realized that both are required or the complete picture.

lying in a region of ℂ𝕄 referred to as the forward tube ℂ𝕄+, this being given by complex

position vectors whose imaginary parts s are past-pointing timelike. In twistor terms such

points are represented by lines lying entirely in ℙ𝕋+, and we see this depicted in figure 8. This

particular arrangement is important for quantum mechanics, because it is a convenient

characterization of the important requirement, for a wave function, of energy positivity (see

[30] and see also the comments at the end of §A3).

Fig. 8:

The geometry relevant to the twistor contour integral for a wave-function. The regions Q1 and Q2 of the text are

the respective complements, within ℙ𝕋+, of the depicted regions 𝒱2 and 𝒱1. Here, the open sets 𝒱1, 𝒱2 provide a

2-set open covering of ℙ𝕋+ and ℛ is their intersection.

C4. Singularity structure for twistor wave-functions

For such expressions to provide non-zero answers, the function f must have appropriate

singularities. The situation of specific interest to us here is the case of a wave-function for a

free massless particle, although these formulae can also be used under many other

circumstances, such as for real solutions of Maxwell’s equations in particular domains. Real

solutions can clearly be obtained from the complex ones described here, by taking the real part,

the equations to be satisfied being linear. Completely general solutions of the equations are

obtained in this way provided that they are analytic. In fact, precursors of these equations were

found long ago, for the Laplace equation by Whittaker [27] in 1903, and for the wave equation

(in 1904) by Bateman [28] who later generalized it for the Maxwell equations in the 1930s, see

[29].

As remarked at the end of §C3, for a wave-function, we require complex solutions of

positive frequency, and here is where the important early motivation for twistor theory referred

to at the end of §A4 was, in a sense, finally satisfied. But, as initially presented, this was only

in a way that seemed somewhat odd. Eventually this apparent oddness was re-interpreted as

something remarkably “natural” when properly understood, with potentially deep implications.

Let us see how this works. First, we take note of the fact, already noted in §C3 that the

family of points of ℂ𝕄 that constitute the sub-region ℂ𝕄+ known as the forward tube,

corresponds to the family of lines that lie entirely in ℙ𝕋+. A complex function , defined on

𝕄, which extends smoothly to a holomorphic function throughout ℂ𝕄+ is indeed of positive

frequency and conversely, positive frequency being a key requirement for a wave-function

[30]. Thus, for our twistor wave-function f, we require “regularity” of an appropriate sort

throughout the region ℙ𝕋+. Yet it would be far too restrictive to demand holomorphicity for f

over the whole of ℙ𝕋+ and, in any case, such a function would simply give the answer zero

when contour integrated. What we seem to need is a function with two separated regions of

singularity on each Riemann sphere (complex projective line) that corresponds to a point in

X

ℂ𝕄+, i.e. to a projective line in ℙ𝕋+, since then we could obtain a non-trivial answer to the

contour integration, the contour being a closed loop on the Riemann sphere that separates the

two regions of singularity on the sphere. The situation is depicted on the right-hand side of

figure 8. This is achieved if the singularities of f are constrained to lie in two disjoint regions

Q1 and Q2 (each region being closed in ℙ𝕋+) so our contour integrations can take place within

the holomorphic region ℛ between them (figure 8). Our twistor wave-function f is thus taken

to be holomorphic throughout the (open) region

ℛ = ℙ𝕋+ – (Q 1 Q 2).

This, appears to be a somewhat odd requirement for the twistor description of such a

fundamental thing as a massless particle’s wave-function. Moreover, the region ℛ is very far

from being invariant under the holomorphic motions of ℙ𝕋+, some of these representing the

non-reflective Poincaré (inhomogeneous Lorentz) motions of Minkowski space 𝕄. Any

particular choice of the region ℛ clearly cannot take precedence over any other such choice

obtained from the original one by such a motion, so there is clearly much non-uniqueness

involved in the choices of ℛ and f in this description. This difficulty looms large if we try to

add two twistor wave-functions which might have incompatible singularity structures.

Linearity is, after all a central feature of quantum mechanics as we currently understand that

subject, so how are we to deal with this problem?

C5. Čech cohomology

The resolution of these puzzling features, leads us to an understanding of what kind of

an entity a twistor wave-function actually “is”. This lies in the notion of Čech sheaf

cohomology. It is not appropriate that we go into much detail, here, but some indication of the

issues involved will be of importance for us. What we find is that the twistor wave-function f

is not really to be viewed as being “just a function” in the ordinary sense, but as representing

an element of “1st cohomology” (actually 1st sheaf cohomology). I shall call such an entity a 1-

function. An ordinary function, in this terminology, would be a 0-function. There are also

higher-order entities referred to as 2-functions, 3-functions, etc., but we shall not need to

consider these here.

An important aspect of 1-functions (or of n-functions, where n>0) is that they are non-

local entities in an essential way (a feature of twistor theory which appears to reflect aspects

of non-locality that occur in quantum mechanics). A good intuitive way of appreciating the

idea of a 1-function is to contemplate the “impossible tribar” depicted in figure 9. Here we

have a picture that for each local region, there is an interpretation provided, of a 3-dimensional

structure that is unambiguous, except for an uncertainty as to its distance from the viewer’s

eye. As we follow around the triangular shape, our interpretation remains consistent (though

with this mild-seeming ambiguity) until we return to our starting point, only to find that it has

actually become inconsistent! The element of 1st cohomology that is expressed by the picture

is a measure of this global inconsistency [31].

Fig. 9:

An impossible “tribar”, as an illustration of the notion of (1st) c0homology. There is an unambiguous interpretation

of each local part, except for an ambiguity as to the distance from the viewer’s eye, but globally this ambiguity

leads to a non-local inconsistency. The measure of this inconsistency is an element of 1st cohomology. Twistor

wave-functions exhibit a similar feature, where the rigidity of analytic continuation replaces the rigidity of a

material body.

How might we assign such a measure to the degree of this impossibility? I shall not go

into full details here, but the idea is to regard the object under consideration—here the tribar—

as being built up from a number of regions (open sets) which together cover the whole object,

but which are “locally trivial” in some appropriate topological (or differential) sense. In the

case of the tribar, we might have a local picture of each vertex, say 𝒱1, 𝒱2, 𝒱3, where the three

pictures overlap pairwise in smaller open regions 𝒱i𝒱j, somewhere along each relevant arm

of the tribar, so that taken together they provide a picture of the entire tribar. On each overlap

region 𝒱i𝒱j, we require some numerical measure Fij which describes the ratio of the

displacement from the eye that needs to be made for the pictures to be considered to match,

and since we need an additive measure we take Fij to be the logarithm of this ratio, and

accordingly the Fij are anti-symmetric (Fji=–Fij), since the ratio goes to its reciprocal when

taken in the opposite order. The triple (F12,F23,F31), taken modulo the particular triples of the

form (H1–H2,H2–H3,H3–H1) where Hi refers to the freedom that is inherent in the interpretation

of each particular vertex picture 𝒱i. The resulting algebraic notion gives us the required

cohomology element, describing the degree of impossibility in the figure. This notion is what

I am calling a 1-function. For further issues see [32].

This is just to give a little flavor of what sort of an entity a 1-function actually is. More

specifically, in the context of twistor theory, we are concerned with complex spaces and

holomorphic functions on them. Thus, in the case of a twistor wave-function there is the

important subtlety, in that the global “impossibility” arises from the “rigidity” of holomorphic

functions rather than that of the solid structures conjured up by the local parts of figure 9. But

let us be a bit more general here, and imagine some complex manifold 𝒦. We shall need a

locally finite open covering 𝒞=(𝒱1,𝒱2,𝒱3,…), of 𝒦. To define a 1-function f, with respect to 𝒞,

we assign a holomorphic function fij on each non-empty pairwise intersection:

fij = –fji is holomorphic on 𝒱i𝒱j,

and on each non-empty triple intersection:

fij + fjk + fki = 0 on 𝒱i𝒱j𝒱k,

where the collection{fij} is taken modulo corresponding collections of the form {hi – hj}, where

each

hi is holomorphic on 𝒱i,

so that two 1-functions are considered to be equal if the difference between their {fij}

representations is of the form {hi – hj}. This defines a 1-function with respect to the particular

covering 𝒞. For the full definition, we would have to take the direct limit for finer and finer

coverings. Fortunately, in the case of complex manifolds, as is being considered here, we are

assured that provided that the sets 𝒱i are of suitable type (e.g. Stein spaces; see [33]) then we

gain nothing from taking such a limit, and the 1-function concept is already with us.

Nevertheless, in order to add two 1-functions defined by different coverings, we do need to

take their common refinement in order to perform this operation, which can be a little

complicated in practice.

In the case of main interest here, namely 𝒦=ℙ𝕋+, it will be adequate for our immediate

purposes here simply to take a 2-set covering of ℙ𝕋+, namely 𝒞={𝒱1,𝒱2}, with open sets given

by the complements, within ℙ𝕋+ of the respective singularity regions Q 2 and Q 1. Then we have

our required covering 𝒞 (not actually with Stein spaces, but that is not of great importance here)

ℙ𝕋 + = 𝒱1 𝒱2, ℛ = 𝒱1 𝒱2;

(see figure 8), just as we had earlier. The family {fij} consists of the single twistor function f,

which, by an abuse of notation I may identify with the 1-function it determines.

This interpretation in terms of 1st cohomology finally fully realized the motivation

described in the final paragraph of §A3. The division described there, of the Riemann sphere

into southern and northern hemispheres by its equator (representing the real numbers), where

positive-frequency functions extend holomorphically into the southern hemisphere and

negative frequency ones into the northern hemisphere is precisely reflected by the division of

ℙ𝕋 into ℙ𝕋 + and ℙ𝕋– by the “equatorial” ℙℕ. The only essential difference, apart from the

increase of dimensionality from the Riemann sphere ℂℙ1 to projective twistor space ℂℙ3, is

that the holomorphic functions (0-functions) on the Riemann sphere are replaced by

holomorphic 1st cohomology elements (1-functions) on projective twistor space.

In the cases of homogeneity 0 or 2 (left-handed electromagnetism or left-handed

linearized gravity, respectively), there are generalizations of the twistor-space contour-integral

expressions that allow one to view the 1-function nature of a twistor function in a different

light, in which non-linearities of general relativity and particle physics begin to play a

significant role. To appreciate this, let us return to the general Čech descriptions given earlier,

where a locally finite open covering 𝒞=(𝒱1,𝒱2,𝒱3,…) of a complex space 𝒦 was considered.

In the specification of a 1-function f, in relation to this covering, we required a family of

holomorphic functions {fij} defined on the non-empty overlaps 𝒱i𝒱j. Here, the functions are

entirely passive, being just “painted on” the space 𝒦. However, we can consider a somewhat

more active role for such a 1-function f, such as (a) specifying the generation fa bundle above

𝒦, or (b) using f to specify the generation of a deformation of 𝒦 itself. In each case, the rules

(see [33]) defining a 1-function are exactly what is needed to fulfil this purpose. However, in

each case, this specification by a 1-function would only be as an infinitesimal generator of the

bundle or deformed space (except for an Abelian group in case (a)) because of non-linearities.

Nevertheless, the general idea expressed in (a) and (b) still holds true; it is just that the linear

nature of a 1-function ceases to hold. In effect, we have a kind of “non-linear 1-function”.

It was in 1977 that Richard Ward introduced the procedure indicated in case (a) above,

first in the situation provided by the (left-handed) Maxwell equations, which allowed

interactions of the field with charged particles to be considered. Almost immediately

afterwards he showed how this procedure could be generalized to the (left-handed) Yang-Mills

equations [34]. This turned out to have considerable importance in the theory of integrable

systems (see, for example, [32], [35]). Shortly before all this, in 1976, procedure (b) had been

introduced [36], to provide a twistorial representation of all conformally complex-Riemannian

4-manifolds which ae anti-self-dual (i.e. �̃�A´B´C´D´=0; see end of §A3). When an additional

simple condition is imposed, this provides not only a (complex) metric but automatically

generates the general anti-self-dual solution of the Einstein vacuum equations, either without

[36] or with a cosmological constant [37].

C6. Infinity twistors and Einstein’s equations

It is a fairly straight-forward procedure to generate the desired deformed twistor spaces

satisfying the required conditions ensuring satisfaction of the Einstein -vacuum equations (

being the cosmological constant, =0 allowed). Basically, what is required is to match

appropriate portions of (non-projective) twistor space, while preserving the Euler operator

𝛶 = Zα 𝜕

𝜕𝑍𝛼

and the 2-form

= Iαβ dZαdZβ

where the anti-symmetrical -infinity twistor Iαβ (and its dual Iαβ) are given by

Iαβ = (Λ

6휀𝐴𝐵 0

0 휀𝐴´𝐵´), 𝐼𝛼𝛽 = (

휀𝐴𝐵 0

0Λ

6휀𝐴´𝐵´

).

We see that 𝐼𝛼𝛽 and Iαβ are both complex conjugates and duals of one another:

Iαβ = 𝐼𝛼𝛽̅̅ ̅̅̅, 𝐼𝛼𝛽 = 𝐼𝛼𝛽̅̅ ̅̅ ,

Iαβ = 12 휀𝛼𝛽𝜌𝜎 𝐼

𝜌𝜎, 𝐼𝛼𝛽 = 12 휀𝛼𝛽𝜌𝜎 𝐼𝜌𝜎,

where 휀𝛼𝛽𝜌𝜎 and 휀𝛼𝛽𝜌𝜎 are Levi–Civita twistors, fixed by their anti-symmetry and

0123=1=0123 in standard twistor coordinates. The preservation of 𝛶 and on the overlaps

where 𝒱i as matched to 𝒱j, is ensured, if we shift infinitesimally along the vector field

𝐼𝛼𝛽 𝜕𝑓𝑖𝑗

𝜕𝑍𝛼

𝜕

𝜕𝑍𝛽 ,

where each fij has homogeneity degree 2 (i.e. 𝛶fij=2fij, which corresponds to helicity 2). We can

imagine exponentiating these infinitesimal deformations to obtain a finite one. In the case of a

2-set covering, we can achieve this explicitly by exponentiating the single function f12, but with

larger numbers of sets, we can encounter difficulties in satisfying the required condition on

triple overlaps. A simpler procedure for satisfying the required condition of preserving 𝛶 and

is to use generating functions, see [36]. This has particular relevance to the procedures of

D4.

It is, however, not at all a direct matter to obtain the (complex) curved space-time ℳℂ

from the deformed twistor space 𝒯, according to this construction. The points of ℳℂ

correspond to “lines” in ℙ𝒯, that are completed Riemann spheres, stretching across from one

patch to the other, or perhaps several patches if the covering involves more than two. These

Riemann spheres are not easy to locate, in a general way, since they are determined by the

global requirement that they be compact holomorphic curves within ℙ𝒯 of spherical topology

(and belonging to the correct topological family). The very existence of these “lines”, as I shall

call the, together with the fact that they belong to a 4-parameter family of such lines (provided

that the deformation from ℙ𝕋 +, or from some other appropriate part of ℙ𝕋, is not too drastic),

depends upon key theorems by Kodaira and Kodaira-Spencer (see [38], [39]). The space whose

points represent these lines is the required complex 4-manifold ℳℂ. Its complex conformal

structure comes about from the simple fact that meeting lines in ℙ𝒯 correspond to null separated

points in ℳℂ, and the definition of its metric scaling comes about through use of the form .

With this complex metric, the complex 4-manifold automatically satisfies the Einstein -

vacuum equations, and the construction provides the general anti-self-dual solution. This

procedure has become known as the “non-linear graviton construction” [36]2, [37]. It has found

numerous applications in differential geometry [32], [35].

At this point, it is worth emphasizing an essential but unusual feature of the non-linear

graviton construction. This is that the “curvature” in the deformed twistor space (encoding

ℳℂ’s actual curvature) is not local, in the sense that a small-enough neighbourhood of any

point in the deformed twistor space is identical in structure with that of ordinary flat twistor

space 𝕋 (with the given assigned to it). The “curvature” in the deformed twistor space is a

non-local feature of the space ℙ𝒯, but in the construction of the “space-time” manifold ℳℂ, we

consequently find genuine local curvature in the normal sense (Riemann curvature, Weyl

curvature).

If we are to regard twistor theory as providing an overall approach to basic physics,

however, then we must face up to the fundamental obstruction to progress that as confronted

the phis programme for some four decades, namely what has become known as the “googly

problem” (an apposite term borrowed from the game of cricket, for a ball bowled with a right-

handed spin about its direction of motion, but bowled with an action that would appear to be

delivering a left-handed spin). This comes about from the very nature of the construction. The

points of ℙ𝒯 have an interpretation within ℳℂ as what are called “α-surfaces” (totally null self-

dual complex 2-surfaces) [36], and there would have to be a 1-complex-parameter family of

such surfaces through each point of ℳℂ (corresponding to the 1-parameter family of points on

2 The historical point must be made here that a key input to the development of the non-linear graviton construction

was the introduction, in 1976, by Ezra T. Newman, of his notion of the ℋ-space, for an asymptotically flat space-

time, as described initially in [40]; for more detail, see [41]. For some fascinating new developments of these

ideas, relating ℋ-space to equations of motion, see [41].

each line of ℙ𝒯). This would imply �̃�A´B´C´D´=0, i.e. ℳℂ being conformally anti-self-dual. Thus,

it is the very existence of points of the space ℙ𝒯 that implies the existence of these troublesome α-

surfaces. This illustrates the fundamental underlying difficulty of the googly problem.

Nevertheless, it is clear that if twistor theory is to have any hope of providing a basis for

fundamental physics, there needs to be a way around this problem. Many ideas for addressing it have

been made over the years, often resorting to examining the twistor structure at infinity, where

the geometry is simpler than that at finite regions (see, for example, [42]), but none has been

able to achieve very much. The most successful approach has been that of ambitwistors [43],

[44], that is based on complex null geodesics, modelled on twistor, dual twistor pairs (Zα,Wβ),

subject to ZαWα=0. This enables complex-Riemannian 4-manifolds to be studied in relation to

twistor-type ideas, and the Einstein vacuum equations to be examined in this light [43]. But it

does not follow the twistor route of “non-linearizing” the 1-function description of quantum

wave functions, where left- and right-handed helicities can be combined together to describe

gravitational interactions and classical space-times in a way that directly relates to twistor wave

functions.

It should be mentioned that even without progress on the googly problem twistor ideas

have found many significant applications, mainly in pure mathematics (most particularly in

certain areas of differential geometry, representation theory, and integrable systems (see, in

particular, [32], [35]), but, relatively recently, also in physics, where great simplifications to

calculations in high-energy scatterings have been obtained. The main impetus to this relatively

recent work came from Edward Witten [45], who introduced several novel ideas, partly based

on earlier work in publications by others [46], [47], [48], [49], [50], [51], [52], [53]), and which

subsequently stimulated a great deal of further activity, e.g. [54], [55], [56], [57], [58], [59],

[60], [61], which allowed twistor methods to help in enormously reducing the calculations of

scattering amplitudes at very high energies (where all particles involved could be considered

to be massless). Nevertheless, despite all this impressive work, relatively little of the deeper

results of twistor theory has been incorporated, and it is my view that these are needed for

twistor theory to realize its aims to become an underlying scheme for fundamental physics

generally. In Part D, I outline what I now believe to be the appropriate direction for progress

to be made in this regard.

Part D: Palatial Twistor Theory

D1. Basic ideas of palatial twistor theory

As has just been remarked upon (in §C6 above), it is the seeming need to have an

unambiguous notion of a point in twistor space that appears to drive us inevitably to this

incomplete and lop-sided approach to space-time in twistor theory, that an unresolved googly

problem presents us with. It is fortunate, therefore, that there is a novel approach to generalizing

the non-linear graviton construction, so that both helicities can be accommodated within the

same general framework, and that classical conformal Lorentzian space-times should also

come under the same umbrella. To understand the basic idea, let us first return to the procedure

that we considered earlier, in the non-linear graviton construction of §C6, where to produce a

suitably deformed twistor space, we “glue together” pieces of complex manifold, preserving

the complex structure from patch to patch. The notion of “complex structure” can be

encapsulated in terms of the algebra of holomorphic functions on each patch—or, technically,

the “sheaf” of such functions, where we require holomorphicity throughout a small

neighbourhood of each point. Now, we saw from the above that the basic conundrum was the

existence of the actual points in each patch, since it was the interpretation of the points in ℙ𝒯

that gave rise to the unwanted α-surfaces. Thus, it would seem, we somehow need to find a

way of matching the sheaves of algebras of holomorphic functions from one patch to another,

without actually having “patches” that, in the ordinary sense, would consist of individual

points. This will not do as it stands, however, because the function algebras already “know”

the points, this being true so long as the algebras are, like the algebra of holomorphic functions

on a region of twistor space, commutative.

This suggests that we take, instead, a holomorphic algebra that is non-commutative!3

As we have seen in §C1, there is a natural non-commutative algebra in twistor theory, namely

that generated (via complex linear combinations and products—these basic operations being

allowed to be repeated many times—and the taking of appropriate limits) from the operators

(see C2):

Zα × and – 𝜕

𝜕𝑍𝛼 .

I shall refer to this algebra as 𝔸, the basic quantum twistor algebra for Minkowski space 𝕄. The

idea is that, in some sense, we could patch together two (or more) “sub-regions” of the algebra

𝔸, analogous to the 𝒱1 and 𝒱2 of the non-linear graviton construction that, in an appropriate

sense “cover” the entire algebraic structure of interest. This “patched” algebra 𝒜 is then to

represent some open portion of the complex(ified) space-time ℳℂ.

The essential idea is that the algebra 𝔸 can be thought of as a system of complex linear

operators acting on holomorphic functions defined locally on some complex space which,

initially, we think of as twistor space 𝕋. In Dirac’s quantum-mechanical terminology [25], 𝕋

is a “ket” space for the algebra 𝔸 of quantum operators. We are to think of 𝔸 as an abstract

algebra that is not dependent upon this particular realization. For example, the same 𝔸 could

also be thought of as the space of complex linear operators acting on holomorphic functions on

3 I am very grateful to Michael Atiyah for making me aware of this important requirement, in a brief conversation

in 2013 (which happened to be at an occasion in Buckingham Palace—hence the name).

the dual twistor space 𝕋*, where the respective operators above would (as displayed in §C2)

now be

𝜕

𝜕𝑊𝛼 and Wα ×

which satisfy the same commutation rules as before. Thus, in Dirac’s terminology, 𝕋* would

be an alternative ket space for 𝔸.

A quantum-mechanical way of thinking about this would be to assert that the

commuting complex parameters Z0, Z1, Z2, and Z3 constitute what we may call a complete set

of commuting operators, in the sense that they, together with their respective partial derivatives

∂⁄∂Z0, ∂⁄∂Z1, ∂⁄∂Z2, and ∂⁄∂Z3, generate, in an appropriate sense, the algebra 𝔸. A different

representation of the same algebra 𝔸 would be obtained if we chose, instead, the complete set

of commuting operators W0, W1, W2, and W3, where the formal component replacements Wα⇝–

∂⁄∂Zα and, accordingly, ∂⁄∂Wα⇝Zα are made. In this sense, 𝕋 and 𝕋* would provide alternative

ket spaces for the same algebra 𝔸. There would be many other possible choices of ket space

for the same algebra 𝔸.

In order to obtain something analogous to the non-linear graviton construction we

would seem to have to involve something analogous to a connected open subset of such a ket

space 𝕋. The general idea would be to patch together various different such “ket patches”, by

analogy with the patching together of different open regions in a locally finite covering

𝒞=(𝒱1,𝒱2,𝒱3,…), if we are to build up a non-trivial (complex) manifold. In an appropriate

sense, the corresponding algebras 𝔸1, 𝔸2, 𝔸3,… would have to “agree” on overlaps (locally

isomorphic on the overlaps, in some suitable sense.), but the ket spaces would generally differ

from patch to patch, so that the “patched-up” algebra 𝒜 that we end up would have no overall

ket space, and would therefore differ from 𝔸, though agreeing with it in a local sense (see [62]

and [63] for earlier tentative descriptions of this idea).

However, there are various thorny issues that need to be faced, with regard to this sort

of “patching” if we are asking for a duly rigorous picture. We might, for example, consider

some sub-region 𝒳 of twistor space 𝕋, which we propose to use as a ket-space “patch”. The

operator exp(Aα∂⁄∂Zα) for constant Aα, in particular, would appear to be harmless enough, but

it could present us with difficulties, as a candidate for membership of the algebra whose ket

space is to be 𝒳. A problematic issue is that a holomorphic function f, defined on 𝒳, whose

analytic continuation from inside 𝒳 to a point displaced by the vector Aα to a somewhere

outside 𝒳, where this analytic continuation of f becomes singular, would exclude exp(Aα∂⁄∂Zα)

from membership of the algebra, since its action on f would be singular. On the other hand, the

operation of multiplication by exp(AαWα), on a ket space that is any sub-region of 𝕋* would be

completely harmless. Such issues need to be better understood for a properly rigorous picture

of this intended procedure to be obtained.

It is clear from all this that there is a considerable vagueness in this proposal, as put

forward above. Most particularly, we do not have a clear notion of topological issues, such as

“local” and “open set”, when it comes to these algebras. These difficult issues are not properly

resolved as things stand, and in the following sections I shall adopt a policy of providing

explicit procedures for the needed patching without worrying about topological issues,

complete riguor, and so forth, leaving such matters to later consideration. Nevertheless, I

believe that we can go some way towards addressing these topological matters if we, in a sense

“stay close to the space ℙ𝒩”, where ℙ𝒩 is the space whose points represent null geodesics in

the space-time ℳ that is intended that we are describing in twistor terms. This will be explored

in the next section.

D2. The spaces of momentum-scaled and spinor-scaled rays

In accordance with this, let us indeed explore the ray-space ℙ𝒩, whose points represent

individual rays in a space-time ℳ, where ℳ is taken to be a smooth time-oriented Lorentzian

globally hyperbolic 4-maifold. Thus ℙ𝒩 plays the same role for ℳ as does ℙℕ for Minkowski

space 𝕄. The global hyperbolicity of ℳ (equivalent to ℳ containing some global spacelike

3-surface which can act as a Cauchy hypersurface for physical fields within ℳ; see [64])

ensures that ℙ𝒩 is Hausdorff and that ℳ contains no causal anomalies such as closed or

“almost closed” rays [64]. We are to imagine that our proposed “palatial space” ℙ𝒯 can, in

some appropriate sense, be viewed as being some sort of “extension” of ℙ𝒩, to include non-

null twistors, though not in any direct sense as an ordinary manifold.

In addition, we shall be interested not only in the 5-manifold ℙ𝒩 of rays in ℳ, but also

in the 6-manifold 𝕡𝒩 of momentum-scaled rays in ℳ. Thus, each point of 𝕡𝒩 represents a

ray together with a momentum co-vector pa which is parallel-propagated along , the future-

null vector pa being tangent to . We shall be interested also in the smooth Hausdorff 7-

manifold 𝒩 representing the spinor-scaled rays , where, in addition to the momentum scaling

pa, we attach a spinor phase provided by the flag plane of 𝜋𝐴′, where

pa = �̅�𝐴𝜋𝐴′,

providing a spinor scaling for (see §B2), these requirements being all conformally invariant

(see [58] Chapter 7). Importantly, this is just the arrangement of spaces required for the

procedure of geometric quantization as described (explicitly, in relation to twistor theory) in

Woodhouse’s book [65], and we shall be seeing the significance for us of some of the ideas of

geometric quantization in the next section, §D3. Explicitly: the 6-space 𝕡𝒩 is a symplectic

manifold, and 𝒩 is a circle bundle over 𝕡𝒩, the circles given by the phase freedom in 𝜋𝐴′.

I shall write the symplectic 2-form of the space 𝕡𝒩as

= i dZαd�̅�𝛼

in anticipation of a role for twistor notation for 𝒩, where we take note of the fact that in the

case of flat twistor space 𝕋 we then find

= dpadxa

(by a brief calculation in the notation of §B2 for twistors in 𝕄). This is the standard symplectic

2-fprm for the cotangent bundle of a manifold, where in this case we are thinking of the

cotangent bundle of 𝕄, symplectically reduced by the Hamiltonian pa pa where we take pap

a=0,

so that the above indeed gives us as he symplectic structure of 𝕡ℕ (i.e. of 𝕡𝒩 when ℳ=

𝕄). See [18], [44] and [64] for details. This suggests that the twistor expression above, for the

2-form might perhaps also have meaning in the case of general Lorentzian-conformal 4-

manifold ℳ. Indeed, a certain justification for this twistorial expression for is given in [18],

§7.4,see fig 7-4, particularly.)

Up to this point, we have been fixing attention on what will be regarded as the region

Zα �̅�𝛼 = 0,

in other words, the actual geometrically defined circle bundle 𝒩 over the symplectic 𝕡𝒩. We

are now going to try to think of this region as somehow extendible as a manifold to where

Zα�̅�𝛼≠0, but there cannot in general be any unique way of doing this. Nevertheless, there are

indeed many way of doing it locally. The picture I am presenting is that we apply the ordinary

notion of a locally finite open covering to ℙ𝒩, this extending to 𝒩, by virtue of the spinor

scaling, as described above. Let (ℙ𝒩1, ℙ𝒩2, …) be such an open covering of ℙ𝒩 extending,

accordingly, to a locally finite open covering (𝒩1,𝒩2,…) of 𝒩-{0}, where we shall require

that in some sense—as yet to be specified more precisely—the individual spaces 𝒩k, together

with their intersections 𝒩i𝒩j, are appropriately “simple” enough that the requirements below

can be satisfied.

Whereas the patchings of these 𝒩-regions are to be pointwise, in the ordinary sense,

this cannot be expected to hold fur the proposed “extensions” to Zα�̅�𝛼≠0. Instead, it will be that

there is a (quantum) twistor algebra 𝔸𝑗 defined for each patch 𝒩j, and it is these algebras that

we shall require match on the overlaps between patches. A key issue is that we cannot

necessarily expect the ket spaces for these algebras will match. For if they did, we would have

a matching of actual spaces, which is just what we wish to avoid, for consistency with the

aspirations expressed in §D1.

We shall be seeing shortly (in §B3) how the procedures of geometric quantization allow

us, in a general way, to construct and match the required algebras 𝔸𝑗, but before proceeding to

this, we shall need a little more about how twistor ideas and notation can be used to describe

the needed structure of these patches. For this, I shall ask that the geometric structure of 𝒩 be

analytic. This may not be essential, but it makes descriptions easier. Analyticity allows us to

complexify 𝒩 to a complex 7-manifold ℂ𝒩 (a complex “thickening” 𝒩 which contains 𝒩 as

a real submanifold). Correspondingly, there will also be complex manifolds ℂℙ𝒩 and ℂ𝕡𝒩,

where ℂ𝕡𝒩 is complex-symplectic and where ℂ𝒩 may be regarded as a holomorphic bundle

over ℂ𝕡𝒩 whose fibre is an open annular region containing the unit circle in the complex

(Wessel) plane.

We are to choose the open regions ℂℙ𝒩j small (i.e. “simple”) enough so that there can

be no obstruction to mapping each ℂ𝒩j holomorphically to a subregion of the 7-complex-

dimensional complexification ℂℕ of ℕ, in a way that preserves the bundle over symplectic

structure of each (there being no local notion of “curvature” for these local structures which

could prevent this). This will allow us to use the ordinary twistor descriptions of Part B to

describe the geometry and algebra of each ℂ𝒩j, as inherited from standard twistor space 𝕋.

This mapping will be far from unique, of course, but the very freedom that is allowed by this

non-uniqueness is an important issue for the construction.

Let us fix attention on one particular ℂ𝒩j, and consider two alternative such twistor

descriptions, which I denote by

(Z, Wα) and (𝒵𝛼, 𝒲𝛼).

We take Z and Wα to be independent twistors and dual twistors and, likewise, we take 𝒵𝛼 and

𝒲𝛼 to be independent. The idea is that whereas the descriptions to be given below should then

lead us to the description of a complex space-time ℳℂ, we can then specialize them, in a way

that will be described in §D4, so as to describe real-Lorentzian space-times ℳ, by being able

to revert to the required Hermiticity:

Wα = �̅�α and 𝒲α = �̅�α.

In the general picture, we are to regard one of these twistor descriptions (Z, Wα) as holding on

one patch, and the other description (𝒵𝛼, 𝒲𝛼) as holding on another which overlaps it. We are

interested in the quantum twistor algebras in each system and how to match these algebras

from patch to patch.

D3. A palatial role for geometric quantization

The one piece of clear geometry that we do wish to match between the two systems is

the symplectic structure given by :

= i dZαdWα = i d𝒵𝛼d𝒲𝛼.

but even this is geometrically meaningful only on the region ℂ𝒩. Outside this region, there is

an arbitrariness involved in the extensions away from ℂ𝒩. Nevertheless, I shall demand that

the above relation continues to be maintained outside ℂ𝒩, this still allows a very considerable

freedom in the extension. In the picture that I am presenting, it will be legitimate to regard the

8-complex-dimensional space of pairs (Z,Wα) to be identifiable, in some local region of each,

with the 8-complex-dimensional of pairs (𝒵𝛼,𝒲𝛼), but any such identification is not to be

regarded as meaningful in itself. It is only a means to an end, namely to identify the allowable

quantum twistor algebras that are to be abstracted from this framework.

Here is where the procedures of geometric quantization come into play—though,

technically, it is only the more primitive procedure of geometric “pre-quantization” that will

be called upon here. The idea will be that we require a bundle-connection on ℂ𝒩, determined

by a 1-form Φ, whose curvature is the 2-form (apart from the factor of i introduced here).

The various possible such connections will give the different possible realizations of the

quantum twistor algebra. These different possibilities come about from the different ways that

the 2-form is expressed as the exterior derivative of a 2-form Φ (here with a factor of i):

= idΦ.

The bundle connection is then given, in the (Z,Wα) system by

(𝜕

𝜕𝑍𝛼 + Pα ,

𝜕

𝜕𝑊𝛼 + Qα

)

where

Φ = Pα dZα + Qα dWα.

Here,

Pα = – 𝜕𝐸

𝜕𝑍𝛼 and Qα =

𝜕𝐹

𝜕𝑊𝛼, where E + F = Zα Wα

,

E and F being (holomorphic) functions of Zα and Wα.

Using this bundle connection, we can realize the operations of the non-commutative

quantum twistor algebra 𝔸. In the description given at the beginnings of §C1 and §C2 (with

ℏ =1), we get the canonical case where the commutation relations can be written:

[Z, 𝑍𝛽] = 0, [–𝜕

𝜕𝑍𝛼, –

𝜕

𝜕𝑍𝛽] = 0, [Z, –

𝜕

𝜕𝑍𝛽] = 𝛿𝛽

𝛼

and we can directly realize these in our bundle connection if we take Pα=0, Qα=Z (i.e. E=0,

F=ZαWα), so that our connection is represented by

(𝜕

𝜕𝑍𝛼, 𝜕

𝜕𝑊𝛼 + Zα

).

Here we take our ket space to be given by holomorphic functions of Z, where Wα is to be taken

constant, so that the first term in the above gives us /Z and the second gives us Z. The same

algebra, but taken with functions of the dual twistors as the ket space would be obtained by

taking Pα= –Wα, Q

α=0 (i.e. E= ZαWα, F=0), so that our connection is represented by

(𝜕

𝜕𝑍𝛼 –Wα, 𝜕

𝜕𝑊𝛼 ).

where the commutation relations can be written:

[𝜕

𝜕𝑊𝛼,

𝜕

𝜕𝑊𝛽] = 0, [Wα, Wβ] = 0, [

𝜕

𝜕𝑊𝛼, Wβ] = 𝛿𝛽

𝛼

and we can directly realize these in our bundle connection by taking Pα= –Wα, Q

α=0 (i.e. F=0,

G=0), so that our connection is represented by

(𝜕

𝜕𝑍𝛼 – Wα, 𝜕

𝜕𝑊𝛼 ).

Now, we can take our ket space to be given by holomorphic functions of dual twistors Wα,

where we take Z to be constant.

Clearly, by making other ways of splitting ZαWα, into a sum of functions E and F we

can arrive at many different realizations of the algebra 𝔸, at least in some local sense. Yet, he

meaning of the word “local”, in this context is clearly something that also needs attention, but,

as indicated in§D1, I am here leaving aside the thorny issues as are raised by topology and

related matters, concentrating primarily on purely formal matters. Nevertheless, such

topological issues must be dealt with at some stage, in order to address the important matters

corresponding to those treated §C6. In relation to this, we need to recall that the considerations

of this section, following the initial paragraph, have been expressed entirely in the (Z,Wα)

system, whereas in order to express the relations in different patches, we need to relate the

(Z,Wα) system of one patch to the (𝒵𝛼,𝒲𝛼) system of another. As things stand, in addition to

topological matters, the actual carrying out of the transformations involved would be likely to

get exceedingly complicated in practical calculations and generally unilluminating. In the

following section I shall show how these twistor transformations between patches can be

greatly facilitated by means of generating functions.

D4. Palatial generating functions and Einstein’s equations

We have seen in §D3 that although the patching together of different regions is to be

understood in terms of the quantum twistor algebras assigned to the various regions, the

construction of these algebras can apparently be achieved in a directly geometrical way. Most

particularly, the relation between one patch and another patch overlapping it can be described

in terms of respective complex symplectic structures that we may assign to each patch as

defined by the different (Z,Wα) and (𝒵𝛼,𝒲𝛼) systems with the same symplectic 2-form . We

want the s to match from path to patch, but the 1-forms Φ would be expected to differ, as

would the bundle connection that Φ defines. This connection provides the needed algebra 𝔸𝑗for

the jth patch and the specific (Z,Wα) or (𝒵𝛼,𝒲𝛼) system that is used to define this algebra,

like a coordinate system in ordinary manifold construction, would not be taken to have

significance.

There will be no loss of generality if we adopt the convention that we choose the ket-

space description for that algebra to be the space of holomorphic functions in the Z variables

rather than the Wα variables (and the 𝒵𝛼 variables rather than the 𝒲𝛼 variables, etc.). In this

way, we can regard the choice of Φ in the (Z,Wα) system as actually determining the quantum

twistor algebra, together with its ket space. But the ket space itself would not be part of the

palatial structure, nor would Φ, and certainly not the specific (Z,Wα), since these structures are

not determined by the algebra. All we ask is that the algebras agree from patch to patch, being

the same on the overlaps.

Yet, it needs to be pointed out that, in all this, I am being deliberately a bit vague in

using terms like “the same as” rather than “isomorphic to”, for the reason that I am not at all

sure what the precise term should be. The global natures of the algebras might be different in

situations where in some more local sense the structure of the algebras ought to be judged as

“the same”. Again, this raises an issue of rigour that I am leaving aside in this article.

The most explicit way to relate one (Z,Wα) system to another one (𝒵𝛼,𝒲𝛼) while

preserving is by means of a generating function, which chooses one set of half the variables

from one system and the other half from the other one:

G(Z,𝒲𝛼).

The remaining variables are then provided by

𝒵𝛼 = 𝜕𝐺

𝜕𝒲𝛼 and 𝑊𝛼 =

𝜕𝐺

𝜕𝑍𝛼

which by direct calculation immediately gives the required

= i dZαdWα = i d𝒵𝛼d𝒲𝛼.

For the purposes of palatial twistor theory, we require this to have a total homogeneity of 2 in

all its variables. In terms of Euler homogeneity operators, this can be expressed as

(Zα 𝜕

𝜕𝑍𝛼 + 𝒲𝛼 𝜕

𝜕𝒲𝛼) G = 2 G

from which, we obtain

G = 12 (ZWα + 𝒵𝛼𝒲𝛼).

Having an explicit expression for the patching, in terms of the given generating

function, and therefore of the twistor algebras and their respective ket spaces, we can envisage

piecing together an entire quantum twistor algebra 𝒜 for a complex space-time ℳℂ. As was

envisaged in §D1. But how are we to identify the points in ℳℂ? From general considerations,

it can be seen that the points ought to correspond to completely commutative 4-dimensional

sub-algebras of 𝒜, although this does not appear to be a sufficient characterization. Yet, in this

kind of way, the notion of points being null separated would have a simple interpretation in

terms of sub-algebras so that the identification of ℳℂ as a complex-conformal manifold ought

to be derived from the structure of the algebra. Clearly much clarification is needed, and not

the least of these problems would appear to be a need for theorems analogues to those of the

Kodaira and Kodaira-Spencer [38], [39] that were so important for the original non-linear

graviton construction of §C6.

Even if all this works out well, there would remain at least four important issues for

this programme to have importance for physics: (1) How do we incorporate the conformal

scaling that will actually provide a space-time metric? (2) How do we express Einstein’s

equations for this metric? (3) how do we ensure that we obtain a real-Lorentzian space-time

ℳ, rather than just a complex one ℳℂ? (4) How do we generalize this to apply to the Ward

construction for Yang-Mills fields [34], and hence to particle physics?

It is perhaps very remarkable that it appears to be possible to address all three of (1),

(2), and (3) with a single generating function of a particular type, which I shall come to very

shortly. With regard to (4), I see no reason why these palatial procedures should not apply also

to the Ward construction and perhaps give some new insights into particle physics. However

for this to be realistic, one needs to understand the proper way to introduce mass. Earlier

proposals for this involved functions of several twistors [20], [22], [23], and it might be well

worth while to re-open these discussions in the light of the above palatial ideas. It would be

interesting to see whether this leads to any new insights.

To end this article I provide a method, using a curious kind of generating function,

which preserves not only the 2-form , but also another 2-form

= 𝑑𝑍𝛼𝑍𝛽𝐼𝛼𝛽 + d𝑊𝛼d𝑊𝛽𝐼𝛼𝛽

where 𝐼𝛼𝛽 and 𝐼𝛼𝛽are the -infinity twistors introduced in §C6. From general considerations

of twistor theory, it can be inferred that the preservation of defines for us not only a metric

scaling (over and above the conformal structure that is inherent in the twistor formalism), but

also the Einstein -vacuum equations (Ricci scalar equals ), although I do not have a direct

argument for this that can be presented succinctly here. Bearing this in mind, If we can find a

generating function that preserves both and , then our palatial procedure should provide

us with the general ℳℂ satisfying the -vacuum equations. Somewhat remarkably this can be

achieved by a generating function

(Z0, W1, Z3, W2; 𝒲0, 𝒵1, 𝒲3, 𝒵2) ,

which is homogeneous of total degree 2, where

= i √Λ

6

and we impose the symmetry that is unchanged if

the 1st and 2nd entries are interchanged, together with the 3rd and 4th being interchanged

or else

the 5th and 6th entries are interchanged together with the 7th and 8th being interchanged.

This should give us a general ℳℂ satisfying the -vacuum equations. If, in addition, we impose

the Hermiticty relation that becomes its complex conjugate under the complete reversal of

the order of the first 4 entries, together with the complete reversal or the order of the last four

entries, then we should obtain a real Lorentzian ℳ satisfying the -vacuum equations. If

desired, we can use first to obtain the required relation between the (Z,Wα) and (𝒵𝛼,𝒲𝛼)

systems and then reconstruct the original type of generating function G used earlier, from the

relation G=1

2(ZWα+𝒵𝛼𝒲𝛼) and proceed as before.

It should be re-iterated that although ideas from quantum mechanics are crucially

incorporated into this construction, this is still just classical Einstein general relativity. There

is no role here for the Planck length, for example. To extend these ideas for a quantum gravity

theory, it would appear that the commutation rules of §C1 should be extended to include

𝑍𝛼𝑍𝛽 – 𝑍𝑍 + 𝑊𝜌𝑊𝜎 𝜖𝛼𝛽𝜌𝜎 = ς 𝐼𝛼𝛽,

ς being some constant connected with the Planck length (and we may note that he above

formula is actually symmetric with respect to interchange of Z with Wα because of an

identity involving the Levi-Civita tensor 𝜖𝛼𝛽𝜌𝜎).

Acknowledgement: I am deeply grateful for financial assistance through a

personal endowment from J.P. Moussouris. REFETRENCES

[1] Penrose, R. (1987) On the origins of twistor theory, in Gravitation and Geometry: a volume

in honour of I. Robinson. eds. W. Rindler and A. Trautman (Bibliopolis, Naples).

[2] Greene, B. (1999) The Elegant Universe; Superstrings, Hidden Dimensions, and the Quest

for the Ultimate Theory. (Jonathan Cape, Random House, London).

[3] Penrose, R. (1959) The apparent shape of a relativistically moving sphere. Proc. Camb.

Phil. Soc. 55, 137-9.

[4] Cartan, É. (1966) The Theory of Spinors (Hermann, Paris).

[5] van der Waerden, B.L. (1929) Spinoranalyse, Nachr. Akad. Wiss. Götting. Math.-Physik

Kl.,100-109.

[6] Dirac, P.A.M. (1936) Relativistic wave equations. Proc. Roy. Soc. (Lond.) A155, 447-59.

[7] Payne, W.T. (1952) Elementary spinor theory Am. J. Phys. 20, 253-62.

[8] Penrose, R. and Rindler, W. (1984) Spinors and Space-Time, Vol. 1: Two-Spinor Calculus

and Relativistic Fields (Cambridge University Press, Cambridge).

[9] Penrose, R. (1960) A spinor approach to general relativity, Ann. Phys. (New York) 10, 171-

201.

[10] Witten, L. (1959) Invariants of general relativity and the classification of spaces Phys.

Rev.113, 357-62.

[11] Corson, E.M. (1953) Introduction to Tensors, Spinors and Relativistic Wave Equations

(Blackie, Glasgow).

[12] Penrose, R. (1965) Zero rest-mass fields including gravitation: asymptotic behaviour,

Proc. Roy. Soc. London, A284, 159-203 .

[13] Penrose, R. (1980) Null hypersurface initial data for classical fields of arbitrary spin and

for general relativity. Gen. Rel. Grav. 12, 225-264

[14] Kerr, R.P. (1963) Gravitational field of a spinning mass as an example of algebraically

[15] Boyer, R. H. and Lindquist, R. W. (1967) Maximal Analytic Extension of the Kerr

metric. J. Math.Phys. 8, 265-281.

[16] Newman, E.T., Couch, E., Chinnapared, K., Exton, A., Prakash, A. and Torrence, R.

(1965) Metric of a rotating charged mass, J. Math. Phys. 6, 918-9.

[17] Robinson, I. and Trautman, A. (1962) Some spherical gravitational waves in general

relativity, Proc. Roy. Soc. (London) A265, 463-73.

[18] Penrose, R. and Rindler, W. (1986) Spinors and Space-Time, Vol. 2: Spinor and Twistor

Methods in Space-Time Geometry (Cambridge University Press, Cambridge) .

[19] Bade, W.L. and Jehle, H. (1953) An introduction to spinors Rev. Mod. Phys. 25,714-28.

[20] Penrose, R. (1975) Twistors and particles: an outline, in Quantum Theory and the

Structures of Time and Space, eds. L. Castell, M. Drieschner and C.F. von Weizsäcker

(Carl Hanser Verlag, Munich).

[21] Hughston, L.P. (1979) Twistors and Particles, Lecture Notes in Physics No. 97 (Springer-

Verlag, Berlin).

[22] Perjés, Z. (1982) Introduction to twistor particle theory, in Twistor Geometry and Non-

Linear Systems, ed. H.D. Doebner & T.D. Palev (Springer-Verlag, Berlin) pp. 53-72.

[23] Perjés, Z. and Sparling, G.A.J. (1979) The twistor structure of hadrons, in Advances in

Twistor Theory, ed. L.P. Hughston and R.S. Ward (Pitman, San Francisco).

[24] Penrose, R. (1968) Twistor quantization and curved space–time Int. J. Theor. Phys. 1, 61-

99 .

[25] Dirac, P.A.M. (1947) The Principles of Quantum Mechanics. (3rd Edition) (Clarendon

Press, Oxford).

[26] Penrose, R. (1969) Solutions of the zero rest-mass equations, J. Math. Phys. 10, 38-9.

[27] Whittaker, E.T. (1903) On the partial differential equations of mathematical physics.

Math. Ann. 57, 333-55.

[28] Bateman, H. (1904) The solution of partial differential equations by means of definite

integrals. Proc. Lond. Math. Soc. (2) 1, 451-8.

[29] Bateman, H. (1944) Partial Differential Equations of Mathematical Physics. (Dover, New

York).

[30] Streater, R. F. and Wightman A. S. (2000) PCT, Spin and Statistics, and All That. 5th edn.

(Princeton Univrsity Press, Princeton, N.J.)

[31] Penrose, R. (1991) On the cohomology of impossible figures [La cohomologie des figures

impossibles] Structural Topology [Topologie structurale] 17, 11-16.

[32] Huggett, S.A., Mason, L.J., Tod, K.P., Tsou, S.T., and Woodhouse, N.M.J. (1998) The

Geometric Universe: Science, Geometry, and the Work of Roger Penrose, (Oxford

University Press, Oxford).

[33] Gunning, R.C. and Rossi, R. (1965) Analytic Functons of Several Complex Variables

(Prentice-Hall, Englewood Cliffs, New Jersey).

[34] Ward, R.S. (1977) On self-dual gauge fields, Phys. Lett. 61A, 81-2.

[35] Mason, L.J. and Woodhouse, N.M.J. (1996) Integrability, Self-Duality, and Twistor

Theory (Oxford University Press, Oxford).

[36] Penrose, R. (1976) Non-linear gravitons and curved twistor theory Gen. Rel. Grav. 7, 31-

52.

[37] Ward, R.S. (1980) Self-dual space-times with cosmological constant, Comm. Math. Phys.

78, 1-17.

[38] Kodaira, K. (1963) On stability of compact submanifolds of complex manifolds. Am. J.

Math. 85, 79-94.

[39] Kodaira, K. and Spencer, D.C. (1958) On deformations of complex analytic structures I,

II, Ann. Math. 67, 328-401, 403-466.

[40] Newman, E.T. (1976) Heaven and its properties Gen. Rel. Grav. 7, 107-11.

[41] Newman, E.T. (1979) Deformed twistor space and H-space. In Complex Manifold

Techniques in Theoretical Physics, eds. D.E.Lerner and P.D.Sommers (Pitman, San

Francisco) pp.154-65.

[42] Penrose, R. (2001) A new angle on the googly graviton. In Further Advances in Twistor

Theory, Vol.III: Curved Twistor Spaces Chapman& Hall/CRC Research Notes in

Mathematics 424, eds. L.J. Mason, L.P. Hughston, P.Z. Kobak, and K. Pulverer (Chapman

& Hall/CRC, London) 264-269. ISBN 1-58488-047-3

[43] LeBrun, C.R.(1985) Ambi-twistors and Einstein’s equations. Class. and Quantum Grav.

2, 555-63.

[44] LeBrun, C.R. (1990) Twistors, ambitwistors, and conformal gravity, in Twistors in

Mathematical Physics, LMS Lec. note ser. 156, eds T.N.Bailey and R.J.Baston

(Cambridge Univ. Press, Cambridge).

[45] Witten, E. (2004) Perturbative gauge theory as a string theory in twistor space,

Commun.Math.Phys.252; 189-258 . arXiv:hep-th/0312171v2

[46] Penrose, R. (1967) Twistor algebra, J. Math. Phys. 8(2), 345-66.

[47] Penrose, R. and MacCallum, M.A.H. (1972) Twistor theory: an approach to the

quantization of fields and space-time, Phys. Repts. 6C, 241-315.

[48] Hodges,++ A.P. and Huggett, S. (1980) Twistor diagrams, Surveys in High Energy Physics

1, 333-53 .

[49] Hodges, A.P. (1982) Twistor diagrams, Physica 114A, 157-75.

[50] Hodges, A.P. (1985) A twistor approach to the regularization of divergences, Proc. Roy.

Soc. London A397, 341-74.

[51] Parke, S. and Taylor, T. (1986) Amplitude for n-gluon scatterings. Phys. Rev. Lett. 56,

2459.

[52] Nair, V. (1988) A current algebra for some gauge theory amplitudes. Phys. Lett. B214,

215-

[53] Shaw, W.T. and Hughston, L.P. (1990) Twistors and strings. In Twistors in Mathematics

and Physics, LMS Lect. Note Ser. 156, eds. T.N. Bailey and R.J. Baston (Cambridge Univ.

Press, Cambridge).

[54] Hodges, A.P. (2006) Twistor diagrams for all tree amplitudes in gauge theory: a helicity-

independent formalism [arXiv:hep-th/0512336v2]

[55] Hodges, A.P. (2006) Scattering amplitudes for eight gauge fields [arXiv:hep-

th/0603101v1]

[56] Hodges, A. (2009) Eliminating spurious poles from gauge-theoretic amplitudes, arXiv

0905.1473v1.

[57] Hodges, A.P. (2013) …………Nature Physics 9, 205-206, doi:10.1038/nphys2597.

[58] Mason, L.J. and Skinner, D. (2009) Scattering Amplitudes and BCFW Recursion in

Twistor Space. arXiv:0903.2083v3

[59] Mason, L., and Skinner, D. (2009) Dual superconformal invariance, momentum twistors

and grassmannians. ….arXiv:0909.0250v2 [hep-th]

[60] Arkani-Hamed, N., Cachazo, F., Cheung, C. & Kaplan, J. (2010) The S-Matrix in Twistor

Space J. High Energy Phys. 1003, 020 .

[61] Arkani-Hamed, N., Hodges, A., Trnka, J. (2014) Positive amplitudes in the amplituhedron.

qrXiv:1412.8478v1 [hep-th].

[62] Penrose, R. (2015) Palatial twistor theory and the twistor googly problem; in theme issue

‘New geometric concepts in the foundations of physics’. Phil. Trans. R. Soc. (Lond.). A

373: 20140237

[63] Penrose, R. (2016) Towards an Objective Physics of Bell Non-Locality: Palatial Twistor

Theory. In Quantum Nonlocality and Reality (eds. Mary Bell and Shan Gao)

Cambridge Core - History, Philosophy and Foundations of Physics, Chinese Academy of

Sciences, Beijing.(Cambridge University Press, Cambridge).

[64] Geroch, R. (1970) Domain of dependence. J. Math. Phys. 11, 437-.

[65] Woodhouse, N.M.J. (1991) Geometric Quantization, Second edition (Clarendon Press,

Oxford).